Fragility Models

Introduction

Fragility functions describe the probability that a structure will reach or exceed a given damage state as a function of an intensity measure (IM). In this framework, fragility functions are assumed to follow a lognormal distribution, which is commonly adopted in seismic risk assessment due to its ability to capture uncertainty in structural response.

Damage State Definition

Four damage states (DS) are considered for structural fragility:

Structural Damage States

Damage State	Label	Description
DS1	Slight	Minor cracking, onset of yielding
DS2	Moderate	Significant cracking, some spalling
DS3	Extensive	Major structural damage
DS4	Complete	Building at or near collapse

The onset of these damage states is inferred from the maximum interstorey drift ratio (\(\theta_{max}\)) attaining pre-defined thresholds based on the capacity curve.

Damage State Thresholds

DS thresholds for MDOF systems are expressed in terms of maximum interstorey drift ratio:

DS Thresholds

Damage State	Maximum Interstorey Drift Ratio Threshold
Slight (DS1)	\(\theta_{max,DS1} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1}) SD_y / h_i]\)
Moderate (DS2)	\(\theta_{max,DS2} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1})(0.67 SD_y + 0.33 SD_u) / h_i]\)
Extensive (DS3)	\(\theta_{max,DS3} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1})(0.33 SD_y + 0.67 SD_u) / h_i]\)
Complete (DS4)	\(\theta_{max,DS4} = \max_{i=1,...,n_s}[\Gamma_1(\phi_{1,i} - \phi_{1,i-1}) SD_u / h_i]\)

where \(SD_y\) and \(SD_u\) are yield and ultimate spectral displacements, \(\Gamma_1\) is the first-mode participation factor, \(\phi_{1,i}\) is the first-mode shape ordinate, and \(h_i\) is the storey height.

Fragility Function Derivation

Lognormal Fragility Functions

Let \(P(DS \ge ds_i | IM)\) denote the probability that damage state \(ds_i\) is exceeded at a given intensity measure level IM. Under the lognormal assumption:

\[P(DS \ge ds_i | IM) = \Phi\left(\frac{\ln(IM) - \ln(\mu_{DS_i})}{\beta_{DS_i}}\right)\]

where:

• \(\mu_{DS_i}\) is the median intensity measure corresponding to damage state \(ds_i\)

• \(\beta_{DS_i}\) is the total logarithmic standard deviation (dispersion)

• \(\Phi(\cdot)\) denotes the standard normal cumulative distribution function

Total Dispersion

The total dispersion accounts for multiple sources of uncertainty:

\[\beta_{DS_i} = \sqrt{\beta_{r2r}^2 + \beta_{b2b}^2 + \beta_{ds}^2}\]

where:

• \(\beta_{r2r}\) (record-to-record or \(\beta_{EDP|IM}\)) represents variability due to ground motion record-to-record uncertainty

• \(\beta_{b2b}\) (building-to-building or \(\beta_{MDL}\)) captures variability in structural properties across nominally similar buildings

• \(\beta_{ds}\) represents uncertainty associated with damage state threshold definition

A building-to-building dispersion of 0.30 is adopted following FEMA P-58 recommendations.

Collapse Fragility

In addition to the four damage states, a separate collapse fragility function is derived using logistic regression on collapse/non-collapse outcomes from the cloud analysis:

\[P(C | IM) = \frac{1}{1 + \exp[-(\alpha_0 + \alpha_1 \ln(IM))]}\]

The collapse cases are identified when:

• Numerical non-convergence occurs during NLTHA

• The EDP exceeds a collapse threshold \(EDP_C = 1.5 \times \theta_{max,DS4}\)

Interactive Viewer

Use the interactive viewer below to explore fragility functions for different building classes.

Model: GEM

Building Class: -- Select Building Class --

Region: -- Select Region --

IMT: -- Select IMT -- PGA SA(0.3) SA(0.6) SA(1.0)

Occupancy: -- Select Occupancy -- Residential (RES) Commercial (COM) Industrial (IND)

Add to Plot Clear All

X-Axis Max (g): 5.0

Fragility Parameters Summary

The table below shows the fragility function parameters for all building classes using the most efficient intensity measure (IM). Parameters include median values (θ) for each damage state and the total dispersion (β).

The complete fragility parameters are available in the summary file: {model_version}/summaries/structural_fragility_params_efficient_imt.csv

Loading fragility parameters table...

Column Descriptions

Fragility Parameters Table Columns

Column	Description
Building Class	GEM taxonomy string identifying the building class
Efficient IM	Most efficient intensity measure for this building class
θ_DS1 to θ_DS4	Median IM values [g] for damage states DS1 through DS4
β_DS1 to β_DS4	Total logarithmic standard deviation for each damage state

References

• Porter K. et al. (2007). Creating fragility functions for performance-based earthquake engineering. Earthquake Spectra.

• Eads L. et al. (2013). An efficient method for estimating the collapse risk of structures in seismic regions. Earthquake Engineering & Structural Dynamics.

• FEMA (2012). Next-Generation Methodology for Seismic Performance Assessment of Buildings (FEMA P-58). Washington, D.C.